Pippenger and Golumbic [1]} proved for all \(k\) -vertex graphs \(H\)  that \(i(H) \ge \frac{k!}{k^k-k} = (1 + o(1))\frac{k!}{k^k}\)  by construction. Although they use different verbiage, the authors use an iterated, balanced blow-up of a graph \(H\)  as their construction for the lower-bound on \(i(H)\) . A blow-up of a graph \(H = (V(H)=\lbrace v_1,\dots ,v_k\rbrace ,E(H))\)  by replacing each vertex \(v_i\)  with a graph \(H_i\)  so that for any distinct \(i,j \in [k]\) , a pair of vertices \(x \in H_i\)  and \(y \in H_j\)  are adjacent if and only if \(v_iv_j \in E(H)\) . We say that a blow-up is balanced (or nearly balanced) if \(\left||H_i| - |H_j|\right| \le 1\)  for all \(i,j \in [k]\) , and call a blow-up iterated if each \(H_i\)  is itself a blow-up of \(H\) , each part in each \(H_i\)  is a blow-up, etc.
